Application Of High-order Average Vector Field Method In Allen-Cahn Equations

Energy dissipation is a important property for many partial differential equations (PDEs), such as the diffusion equations, the Allen-Cahn equations and the Cahn-Hilliard equations, These equations are familiar to us. They all have the energy dissipation property. The numerical schemes, which can accu-rately calculate the numerical solution of these equations and preserve the corresponding energy dissi-pation property, have essential advantages in simulating the PDEs. This idea comes from the symplectic method, proposed by Feng Kang academician and his stduy team, who is one of the famous computation mathematicians. Domestic and international scholars have proposed the multi-symplectic method for a class of PDEs with the multi-sympectic structure based on the idea of this symplectic method. In recent years, the average vector field (AVF) method of the Hamilton system is proposed, which is an efficient discrete gradient method. The method can exactly preserve the energy conservation of the Hamilton sys-tem. Moreover, the discrete gradient method can also be applied to solve the energy-dissipation PDEs, which can preserve the energy-dissipation property of these PDEs. The second order average vector field method has been widely applied to solve the energy dissipation PDEs, such as the diffusion equations, the Allen-Cahn equations and the Cahn-Hilliard equations.Recently, abroad scholars improved the discrete gradient method and proposed the high-order av-erage vector field method. The high-order average vector field method is applied to solve the ordinary differential equations. At the same time, we have applied the high-order average vector field method to solve the energy-preserving Schrodinger partial differential equations and obtained very good numerical results. At present, few people apply the high-order average vector field method to solve the energy dissipation partial differential equations at home and aborad. We have applied the high order average vector field method to solve the energy dissipation Cahn-Hilliard equations. But the time step length must be very small and the computation time must be short. In this paper, we mainly solve the Allen-Cahn equations by the high-order AVF method. We analyze whether the high-order AVF method can calculate the Allen-Cahn equations accurately with a big time step for long time and whether the energy dissipation property of the Allen-Cahn equations can be preserved for long time. At last, the high-order energy dissipation scheme for the two dimension Cahn-Hilliard equation is also proposed.In chapter 1,we first give the relation between the energy-conservation or energy-dissipation partial equations and their variation equations; In section 2, we give several energy-conservation pratial dif-ferential equations and their corresponding variational equations; In section3, we introduce the discrete gradient method,give the two famous discrete gradient methods:the corresponding average vector field method and the discrete variation method.In chapter 2,we solve the energy-dissipation partial differential Allen-Cahn equations by the high-order AVF method. The first section introduces the historical background of Allen-Cahn equation and proves the dissipation property of its energy function; In section 2, the theory and the computation for-mula of the fourth order average vector field method is given. In section 3, we get the high-order accurate numerical scheme of the Allen-Cahn equation; In section 4, we simulate the Allen-Cahn equation nu-merically and get the evolution behaviors with different initial values. We obtained that the high-order AVF method can preserve the energy dissipation property of Allen-Cahn equation for long timeIn chapter 3, we study the Cahn-Hilliard equation. In section 1, we introduce the pseudo-spectral method for two dimension partial differential equations. We apply the high-order discrete gradient and pseudo-spectral method to solve the two dimension Cahn-Hilliard equation. At last, we get the high-order discrete scheme of the two dimension Cahn-Hilliard equation.